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'''Schur–Weyl duality''' is a mathematical theorem in [[representation theory]] that relates irreducible finite-dimensional representations of the [[general linear group|general linear]] and [[symmetric group|symmetric]] groups. It is named after two pioneers of representation theory of [[Lie group]]s, [[Issai Schur]], who discovered the phenomenon, and [[Hermann Weyl]], who popularized it in his books on [[quantum mechanics]] and [[classical groups]] as a way of classifying representations of [[unitary group|unitary]] and general linear groups.

== Description ==
Schur–Weyl duality forms an archetypical situation in representation theory involving two kinds of [[symmetry]] that determine each other. Consider the [[tensor]] space

: <math> \mathbb{C}^n\otimes\mathbb{C}^n\otimes\cdots\otimes\mathbb{C}^n </math> with ''k'' factors.

The [[symmetric group]] ''S''''k'' on ''k'' letters [[group action|acts]] on this space (on the left) by permuting the factors,

: <math> \sigma(v_1\otimes v_2\otimes\cdots\otimes v_k) = v_{\sigma^{-1}(1)}\otimes v_{\sigma^{-1}(2)}\otimes\cdots\otimes v_{\sigma^{-1}(k)}.</math>

The general linear group ''GL''''n'' of invertible ''n''×''n'' matrices acts on it by the simultaneous [[matrix multiplication]],

: <math> g(v_1\otimes v_2\otimes\cdots\otimes v_k) = gv_1\otimes gv_2\otimes\cdots\otimes gv_k, \quad g\in GL_n. </math>

These two actions [[Equivariant map|commute]], and in its concrete form, the Schur–Weyl duality asserts that under the joint action of the groups ''S''''k'' and ''GL''''n'', the tensor space decomposes into a direct sum of tensor products of irreducible modules for these two groups that determine each other,

: <math> \mathbb{C}^n\otimes\mathbb{C}^n\otimes\cdots\otimes\mathbb{C}^n = \sum_D \pi_k^D\otimes\rho_n^D. </math>

The summands are indexed by the [[Young diagram]]s ''D'' with ''k'' boxes and at most ''n'' rows, and representations <math>\pi_k^D</math> of ''S''''k'' with different ''D'' are mutually non-isomorphic, and the same is true for representations <math>\rho_n^D</math> of ''GL''''n''.

The abstract form of the Schur–Weyl duality asserts that two algebras of operators on the tensor space generated by the actions of ''GL''''n'' and ''S''''k'' are the full mutual [[centralizer (ring theory)|centralizer]]s in the algebra of the endomorphisms <math>\mathrm{End}_\mathbb{C}(\mathbb{C}^n\otimes\mathbb{C}^n\otimes\cdots\otimes\mathbb{C}^n).</math>

== Example ==
Suppose that ''k'' = 2 and ''n'' is greater than one. Then the Schur–Weyl duality is the statement that the space of two-tensors decomposes into symmetric and antisymmetric parts, each of which is an irreducible module for ''GL''''n'':

: <math> \mathbb{C}^n\otimes\mathbb{C}^n = S^2\mathbb{C}^n \oplus \Lambda^2\mathbb{C}^n.</math>

The symmetric group ''S''''2'' consists of two elements and has two irreducible representations, the [[trivial representation]] and the [[sign representation]]. The trivial representation of ''S''2 gives rise to the symmetric tensors, which are invariant (i.e. do not change) under the permutation of the factors, and the sign representation corresponds to the skew-symmetric tensors, which flip the sign.

== References ==
* [[Roger Evans Howe|Roger Howe]], ''Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond''. The Schur lectures (1992) (Tel Aviv), 1–182, Israel Math. Conf. Proc., 8, Bar-Ilan Univ., Ramat Gan, 1995. {{MR|1321638}}
* [[Issai Schur]], ''Über eine Klasse von Matrizen, die sich einer gegebenen Matrix zuordnen lassen''. Dissertation. Berlin. 76 S (1901) JMF 32.0165.04
* [[Issai Schur]], ''Über die rationalen Darstellungen der allgemeinen linearen Gruppe''. Sitzungsberichte Akad. Berlin 1927, 58–75 (1927) JMF 53.0108.05
* [[Hermann Weyl]], ''The Classical Groups. Their Invariants and Representations''. Princeton University Press, Princeton, N.J., 1939. xii+302 pp. {{MR|0000255}}

{{DEFAULTSORT:Schur-Weyl duality}}
[[Category:Representation theory]]
[[Category:Tensors]]

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Constellation model